Spatial Point Processes

Property (iv) is called boundedly finite. So, when we use (iv) instead of (iii), we are interested in simple, boundedly finite spatial point processes with no fixed atoms. When we use (iii), the domain A can be an arbitrary set. When we use (iv), the domain A must have some notion of boundedness. This is no problem when A is a subset of Rd for some d. We just use the usual notion of boundedness. If we were to try to characterize the process by a Bernoulli random variable X(t) at each location t, then property (ii) would just make all the Bernoulli random variables almost surely zero. So that won’t work. Another way to think about the process is that the total number of points N is a nonnegative-integer-valued random variable and conditional on N the locations of the N points are a random vector taking values in An. Yet another way is to think the process as given by random variables N(B) for B ∈ B, where B is a sigma-algebra in A. N(B) is the number of points of the process in the region B. So this is somewhat like our previous